Binomial Distibutions

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Binomial Distibutions. Target Goal: I can determine if the conditions for a binomial random variable are met. I can find the individual and cumulative binomial probabilities using the calculator. 6.3a h.w: pg 381: 61, 65, 66; pg 403: 69, 71, 73. The Binomial Distribution.

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### Binomial Distibutions

Target Goal:

I can determine if the conditions for a binomial random variable are met.

I can find the individual and cumulative binomial probabilities using the calculator.

6.3a

h.w: pg 381: 61, 65, 66; pg 403: 69, 71, 73

The Binomial Distribution

When we are studying situations with two possible outcomes we are interested in a binomial setting.

What are some outcomes with two possible outcomes?

• Toss a coin
• Shoot a free throw
• Have a boy or a girl
Binomial Setting

Suppose the random variable X = the number of successes in n observations.

Then X is a binomial random variable if:

• Binary:there are only two outcomes, success or failure.
• There is a fixed number n of trials.
• The n trials are independent.
If X is a binomial random variable, it is said to have a binomial distribution, and is denoted as

B(n, p)

• n: # of observations
• p: probability of success
A binomial distribution or not?

Blood Types

Parents carry genes O and A blood type. The probability a child gets two “O” genes is 0.25. If there are 5 children in a family, is each birth independent?

• Yes, so X is the count of success

(“O” blood type), and

X is B(5, 0.25)

Dealing cards
• No. If you deal cards without replacement, the next card is affected by the previous. Not independent.
Inspecting Switches
• In a shipment, 10 % of the switches are bad (unknown to the inspector).
• If the engineer takes a SRS of 10 switches from 10,000. The engineer counts X, the number of bad switches.
• Is this Binomial?
• But, if the SRS is 10,000, removing one changes the remaining 9,999 very little.
• When the population is much larger than the sample, we say the distribution is approximately binomial and very close to:
• B(10,0.10)
Activity: A Gaggle of Girls
• How unusual is it for a family to have three girls if the probability of having a boy and a girl is equally likely?
• If success = girl and failure = boy, then p(success) =

0.5.

Define the random variable X as the number of girls.
• We want to simulate families with three children.
• Our goal is to determine the long term relative frequency of a family with 3 girls, P(X=3)
Using the Random Number Table D
• Let the evendigitsrepresent “girl” and the odd digits represent “boy”.
• Each student select their own row, and beginning at that row, read off numbers three at a time.
• Each three digits will constitute one trail.
• Use tally marks to record the results of 40 trails which will then be pooled with the class.
Calculate the relative frequency of the event.

P(X=3) = /40 =

• vs. Class relative frequency:
Using the Calculator
• Using the codes 1 = girl and 0 = boy;
• Enter the command math:prb:randint(0,1,3).
• This command instructs the calculator to randomly pick a whole number from the set {0,1} and do this three times.
• The outcome {0,0,1} represents {boy, boy, girl}.
Continue to press ENTER until you have 40 trails.
• Use a tally mark to record each time a {1,1,1} result.
Calculate the relative frequency of the event.

P(X=3) = /40 =

• vs. class relative frequency:
• Do the results of our simulation come close to the theoretical value for P(X=3) which is 0.125?
• Even quicker…Try
Math:PRB:randBin(#trials,prob,#of simulations) randBin(3,.5,40) store L1
• (2 1 0 3 …); 3 is 3 girls or next,
• Sum(L1=3): count the # of 3 girl possibilities.
• It changes the 3’s to 1(true) and counts.
Finding Binomial Probabilities using the Calculator
• The probability distribution function(p.d.f.) assigns a probability to each value of X.

P(X=2)

• Calculator: TI-83:

2nd:VARS: binompdf (n, p, X)

Cumulative Distributions
• The cumulative distribution function(c.d.f.) calculates the sum of the probabilities up to X.
• P(X≤ 2) = P(X=0) + P(X=1) + P(X = 2)
• Calculator: TI-83:

2nd:VARS: binomcdf (n, p, X)

Example: Glex’s Free Throws
• Over an entire season Glex shoots 75% free throw percentage. She shoots 7/12 and the fans think she was nervous.
• Is this unusual for Glex to shoot so poorly?
Studies of long series found no evidence that free throws are dependent so assume free throws are independent.
Probability of success = 0.75
• Find the probability of making at most 7 free throws.

B(n, p)

B(12, 0.75)

P(X≤ 7) = P(X=0) + P(X=1) + …. + P(X = 7)
• 2nd:VARS(dist):binomcdf (n, p, X)

binomcdf (12, 0.75, 7)

= 0.1576

• Conclusion: Glex will make at most 7 out of 12 free throws about 16% of the time.
Example : Type O Blood Type
• Suppose each child born to John and Katie has probability 0.25of having blood type O. If John and Katie have 5 children, what is the probability that exactly 2 of them have type O blood?
• Binomial distribution?
X: the number of children with type O blood.
• Calculator: TI-83: binompdf(n, p, X)
• Note: binompdf ( 5, 0.25, 0) = 0.2373, finds the P(X=0), none of the children have type O blood.
• TI-89: tistat.binomPdf(n, p, X)
Calculator Procedure:
• Enter values of x; 0, 1, 2, 3, 4, 5 into L1
• Enter the binomial probabilities into L2.
• Highlight L2 and enter 2nd:VARS(dist):

binompdf (5, 0.25, L1).

• Fill in table. (2 min)

P(X=2) = 0.2637

Plot a histogram of the binomial pdf.
• Deselect or delete active functions in Y = window.
• Define Plot1 to be a histogram with Xlist: L1, Freq: L2
• Set the window X[0, 6]1 and Y[0, 1]0.1
Use the TRACE button to inspect the heights of the bars.
• Verify that the sum of the probabilities is 1.

STAT:CALC:1-VAR Stats L2

Fill in the following table of the cumulative distribution function (c.d.f.) for the binomial random variable, X.
• To calculate cumulative probabilities:
• Highlight L3 and enter

2nd:VARS(dist): binomcdf (5, 0.25,L1).

Construct a histogram of the c.d.f.:
• Define Plot1 to be a histogram with Xlist: L1, Freq: L3
• Use the window X[0, 6]1 and Y[0, 1]0.1.
• Use the TRACE button to inspect the heights of the bars.
What do the heights represent?

The cumulative total at each X value.

Compare:
• p.d.f.
• c.d.f.